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Here is a really interesting article on the short tenure enjoyed by most holders of the title of world's oldest person.
Here is a cool graphic from article, showing how old all the recent record holders have been, and how long they've reigned:
As you can see, since 2000 the oldest living person (invariably a woman) has been between the age of 114 and 117, and they've held the title for a short time relative to the 9 years that Jeanne Calment was at the top.
Despite dramatic increases in recent average lifespan, the age of the oldest person alive has remained pretty stable lately:
The average age of the oldest-ever people has increased over the past 40 years from around 112 to around 114.
--Dean